Discrete Energy Conservation Laws Research Articles

On stability and error bounds of an explicit in time higher-order vector compact scheme for the multidimensional wave and acoustic wave equations

We study a three-level explicit in time higher-order vector compact scheme, with additional n sought functions approximating second order non-mixed spatial derivatives of the solution, for an initial-boundary value problem for the n-dimensional wave equation and acoustic wave equation, with the variable speed of sound, n⩾1. We also approximate the solution at the first time level in a similar two-level manner, without using derivatives of the initial data as usual. For the first time, under the CFL-type conditions, new stability bounds in the standard and stronger mesh energy norms and the discrete energy conservation laws are presented, and the corresponding error bounds of the orders 4 and 3.5 are rigorously proved. Generalizations to the cases of the nonuniform meshes in space and time are described. Results of various numerical experiments are also included.

A high-order structure-preserving difference scheme for generalized fractional Schrödinger equation with wave operator

This paper focuses on the construction and analysis of the structure-preserving algorithm for generalized fractional Schrödinger equation with wave operator. A fourth-order energy-conserving difference scheme is developed for the resulting equivalent system based on scalar auxiliary variable approach. The discrete energy conservation law, boundedness and convergence of difference solutions are proved in detail. Numerical experiments are performed to verify our theoretical analysis results.

Higher-order energy-preserving difference scheme for the fourth-order nonlinear strain wave equation

This paper focus on construction of high-order energy-preserving difference scheme for the fourth-order nonlinear strain wave equation with an energy conservation law. This target model is firstly transformed into an equivalent system by using the method of trigonometric scalar auxiliary variables. The resulting equivalent system possess a modified energy conservation law, and a fourth-order difference scheme with analogously discrete energy conservation law is developed based on the resulting equivalent system. The boundedness and convergence of the numerical solutions in the maximum norm are shown. The effectiveness of the difference scheme is verified by several numerical experiments.

Exponentially fitted multisymplectic scheme for conservative Maxwell equations with oscillary solutions.

Aiming at conservative Maxwell equations with periodic oscillatory solutions, we adopt exponentially fitted trapezoidal scheme to approximate the temporal and spatial derivatives. The scheme is a multisymplectic scheme. Under periodic boundary condition, the scheme satisfies two discrete energy conservation laws. The scheme also preserves two discrete divergences. To reduce computation cost, we split the original Maxwell equations into three local one-dimension (LOD) Maxwell equations. Then exponentially fitted trapezoidal scheme, applied to the resulted LOD equations, generates LOD multisymplectic scheme. We prove the unconditional stability and convergence of the LOD multisymplectic scheme. Convergence of numerical dispersion relation is also analyzed. At last, we present two numerical examples with periodic oscillatory solutions to confirm the theoretical analysis. Numerical results indicate that the LOD multisymplectic scheme is efficient, stable and conservative in solving conservative Maxwell equations with oscillatory solutions. In addition, to one-dimension Maxwell equations, we apply least square method and LOD multisymplectic scheme to fit the electric permittivity by using exact solution disturbed with small random errors as measured data. Numerical results of parameter inversion fit well with measured data, which shows that least square method combined with LOD multisymplectic scheme is efficient to estimate the model parameter under small random disturbance.

On higher-order compact ADI schemes for the variable coefficient wave equation

We consider an initial-boundary value problem for the n-dimensional wave equation, n⩾2, with the variable sound speed with the nonhomogeneous Dirichlet boundary conditions. We construct and study three-level in time and compact in space three-point in each spatial direction alternating direction implicit (ADI) schemes having the approximation orders O(ht2+|h|4) and O(ht4+|h|4) on the uniform rectangular mesh. The study includes stability bounds in the strong and weak energy norms, the discrete energy conservation law and the error bound of the order O(ht2+|h|4) for the first scheme as well as a short justification of the approximation order O(ht4+|h|4) for the second scheme. We also present results of numerical experiments.

Arbitrarily high-order energy-preserving schemes for the Camassa-Holm equation

In this paper, we develop a novel class of arbitrarily high-order energy-preserving schemes for the Camassa-Holm equation. With the aid of the invariant energy quadratization approach, the Camassa-Holm equation is first reformulated into an equivalent system, which inherits a quadratic energy. The new system is then discretized by the standard Fourier pseudo-spectral method, which can exactly preserve the semi-discrete energy conservation law. Subsequently, a symplectic Runge-Kutta method such as the Gauss collocation method is applied for the resulting semi-discrete system to arrive at an arbitrarily high-order fully discrete scheme. We prove that the obtained schemes can conserve the discrete energy conservation law. Numerical results are addressed to confirm accuracy and efficiency of the proposed schemes.

Conservative compact finite difference scheme for the N‐coupled nonlinear Klein–Gordon equations

In this article, a compact finite difference method is developed for the periodic initial value problem of the N‐coupled nonlinear Klein–Gordon equations. The present scheme is proved to preserve the total energy in the discrete sense. Due to the difficulty in obtaining the priori estimate from the discrete energy conservation law, the cut‐off function technique is employed to prove the convergence, which shows the new scheme possesses second order accuracy in time and fourth order accuracy in space, respectively. Additionally, several numerical results are reported to confirm our theoretical analysis. Lastly, we apply the reliable method to simulate and study the collisions of solitary waves numerically.

Second-order linear structure-preserving modified finite volume schemes for the regularized long wave equation

In this paper, we develop four energy-preserving algorithms for the regularized long wave (RLW) equation. On the one hand, we combine the discrete variational derivative method (DVDM) in time and the modified finite volume method (mFVM) in space to derive a fully implicit energy-preserving scheme and a linear-implicit conservative scheme. On the other hand, based on the (invariant) energy quadratization technique, we first reformulate the RLW equation to an equivalent form with a quadratic energy functional. Then we discretize the reformulated system by the mFVM in space and the linear-implicit Crank-Nicolson method and the leap-frog method in time, respectively, to arrive at two new linear structure-preserving schemes. All proposed fully discrete schemes are proved to preserve the corresponding discrete energy conservation law. The proposed linear energy-preserving schemes not only possess excellent nonlinear stability, but also are very cheap because only one linear equation system needs to be solved at each time step. Numerical experiments are presented to show the energy conservative property and efficiency of the proposed methods.

Convergence of time-splitting energy-conserved symplectic schemes for 3D Maxwell’s equations

We propose two symplectic and two non-symplectic schemes for 3D Maxwell’s equations based on the exponential operator splitting technique and Fourier pseudo-spectral method. These schemes are efficient and unconditionally stable, and also preserve four discrete energy conservation laws simultaneously. The error estimates of the schemes are obtained by using some special techniques and the energy method. Numerical results confirm the theoretical analysis. The numerical comparison with some existing methods show the good performance of the proposed schemes.

Finite Element Approximations of Wave Maps into Spheres

Three fully discrete finite element methods are developed for approximating wave maps into the sphere based on two different approaches. The first method is an explicit scheme and the numerical solution satisfies the sphere-constraint exactly at every node. The second and third methods are implicit schemes which are based on a penalization approach, their numerical solutions satisfy the sphere-constraint approximately, and the quality of approximations is controlled by a small penalization parameter. Discrete energy conservation laws which mimic the underlying differential conservation law are established, and convergence of all proposed methods is proved. Computational experiments are also provided to validate the proposed methods and to present numerical evidence for possible finite-time blow-ups of the wave maps.

Discrete energy conservation law and reciprocity relationship for one-dimensional wave propagation problems

A mathematical derivation is presentcd to establish the discrete energy conservation law and reciprocity relationship for explaining the insensitive phenomenon round in the previous investigations of a one-dimensional wave propagation problem. It is shown that there do exist certain discretized differential and variational systems which can preserve the energy conservation and reciprocity relationships. This then excludes the use of these relationships in error estimation.